Invariant weakly positive semidefinite kernels with values in topologically ordered ∗-spaces
Instytut Matematyczny PAN
255 - 294
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We consider weakly positive semidefinite kernels valued in ordered ∗-spaces with or without certain topological properties, and investigate their linearisations (Kolmogorov decompositions) as well as their reproducing kernel spaces. The spaces of realisations are of VE (Vector Euclidean) or VH (Vector Hilbert) type, more precisely, vector spaces that possess gramians (vector valued inner products). The main results refer to the case when the kernels are invariant under certain actions of ∗-semigroups and show under which conditions ∗-representations on VE-spaces, or VH-spaces in the topological case, can be obtained. Finally, we show that these results unify most of dilation type results for invariant positive semidefinite kernels with operator values as well as recent results on positive semidefinite maps on ∗-semigroups with values operators from a locally bounded topological vector space to its conjugate Z-dual space, for Z an ordered ∗-space.
Topologically ordered ∗-space VE-space
Weakly positive semidefinite kernel
Published Version (Please cite this version)https://doi.org/10.4064/sm8807-1-2018
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Ay, S.; Gheondea, A. (Elsevier, 2015)We consider positive semidefinite kernels valued in the ∗-algebra of adjointable operators on a VE-space (Vector Euclidean space) and that are invariant under actions of ∗-semigroups. A rather general dilation theorem is ...